Summary

Advertisers constantly thrust quantitative information in our face. Product claims, store enticements, health benefits, and scores of other contexts use short quantitative arguments to catch a reader's eye (and possibly money). This example shows how one can use these ads to bring added content to a quantitative reasoning course.

Learning Goals

Learning goals include

Critical reading of quantitative information,

Understanding of "odds" and probability,

Simple modeling to test assumptions, and

Recognizing misuse or confusing use of language to describe change.

Context for Use

Survival Rates: Could be used at any time in any quantitative reasoning course. It is especially useful in introducing the notion of "odds" or probability. Note that the ad uses the term "odds" although the quantitative information is expressed in terms of a probability.

Automobile Trade-in: Could be used at any time in any quantitative reasoning course. This is a nice example to instill the helpfulness of mental arithmetic or approximation. Can one roughly approximate the range of deductions that is allowable under the advertised claim?

Environmental Claim: Again, this could be used at any time in any quantitative reasoning course. It is especially useful to indicate the confusing language that often surrounds the comparison of two quantities. Students should be taught to be extra careful when reading phrases such as "... percent more than..." or "... percent less than..." as they are often misused or used in confusing or ambiguous ways.

Description and Teaching Materials

Survival Rates

In a large one-page advertisement in the New York Times on October 1, 2008 the following appeared in the space above the fold:

odds of surviving airline crash: 24%

odds of surviving pancreatic cancer: 4%

Study Questions:

What, most likely, is the intent of the advertisers?

How are these two numbers comparable?

How are they not comparable?

What information is missing that would make it easier to compare the likelihood of surviving an airplane crash with the likelihood of surviving pancreatic cancer?

Automobile Trade-in Value

In a mailing from a car dealership it was advertised that the dealership was offering 100% reimbursement for an owner's current vehicle when purchasing a new car. Phrases such as "100% reimbursement...0% Down", "Any customer trading in a 1997-2006 model on a 2008 or 2009 model will receive 100% of the factory full base model MSRP when new (as provided by Black Book)!" "All customers will receive 100% of the base model MSRP in trade toward a new car."

Eventually the fine print yields "The only deductions will be made for mileage ($0.15 - $0.75 per mile depending on make, model, and reconditioning)."

c. According to the calculator, what would it take to reduce your carbon footprint by 54%?

d. Investigate the amount of variation in the calculation of your carbon footprint. That is, how much does your calculated footprint change if you make slight changes in some of your responses? What lifestyle activity appears to have he greatest impact on your carbon footprint?

Teaching Notes and Tips

Survival Rates: This was an advertisement by Cablevision to announce its support of The Lustgarten Foundation's work in pancreatic cancer research. There are several issues that can be brought up in a discussion of this ad:

The ad claims it is reporting the "odds" although odds are used to report a ratio of probabilities. For example, if the probability of an event happening is 20%, then the probability of the event not happening is 80%. Since the ratio of these two probabilities is 20/80 = 1/4, one would say that the odds in favor of the event happening are 1 to 4. Thus, it is unclear if the ad is attempting to report a ratio of probabilities or simply the single probability of surviving a specific event. The two possible interpretations are:

i. The probability of surviving an airline crash is 24%. (And the probability of surviving pancreatic cancer is 4%.)

ii. The odds of surviving an airline crash to not surviving an airline crash is .24. This is equivalent to saying that the probability of surviving an airline crash is approximately 19.4%. (Similarly, the probability of surviving pancreatic cancer is 3.8%.)

Clearly the ad wishes us to compare the 24% with the 4% quantities thereby reaching the conclusion that pancreatic cancer is much more disastrous than an airline crash, which certainly brings to mind horrible images. Assuming these figures are correct, this ad does indeed educate the public on the seriousness of pancreatic cancer. Getting pancreatic cancer is worse than being in an airline crash. However, both of these figures are conditional probabilities. That is, we are told that if one is in an airline crash, the probability of surviving is 24%. The ad does not indicate what the likelihood of being involved in a airline crash is. Or, perhaps more importantly, the ad does not indicate how prevalent pancreatic cancer is in our society. Not trying to diminish the seriousness of pancreatic cancer but simply studying the quantitative information provided, it could be possible that pancreatic cancer is so rare that the low survival rate is not of serious concern.

This ad invites further research on how both the figures were calculated. One might suspect that since air disasters are fairly infrequent, trying to estimate the probability of surviving an airplane crash with real data could be a very sensitive calculation. The timeframe used could dramatically alter the estimation. For example, if one were to use the last ten years of data that happened to exclude a major disaster which occurred eleven years ago could significantly alter the estimate.

Automobile Trade-in: This is a great example to practice basic assumption making as well as quick mental estimations. Quantitative reasoning should be thought of as a habit of mind. If a student is shown this advertisement does he or she take a moment to figure out how much "wiggle room" the automobile dealer has in the "100% reimbursement" promise? Suppose one has a low-mileage older model to trade in. Let's assume one only averages 7,000 miles a year for nine years. This is 63,000 miles. If the dealer sticks to one of the lower values for deduction, say $0.20, this would amount to a $12,600 deduction. A mid-range value of $0.50 is also easy to calculate in one's head: about $31,500. Since the new MSRP of the types of cars sold at this dealership is around $20,000 - $25,000, the dealer is essentially promising nothing.

Environmental Claim: Phrases which attempt to describe percentage increases or decreases are often misleading or ambiguous, especially when one of the figures is greater than 100%. For this example, it is important to investigate if there is a consistent reasonable interpretation of the phrase "... reduces x by y percent...".

If one were to reduce x by 54 percent (think about a store having a 54% discount), one would calculate the new value to be (1- 0.54)x = 0.46x. This same reasoning would imply that to reduce a value x by 300 percent one would calculate (1 - 3.00)x = -2.00x. A calculation which makes no sense when discussing fuel consumption.

What point is this shuttle bus company trying to get across? Sometimes writers try to use the phrase "x times more than" and "x times less than" as inverse or opposite meanings. For example, there is little disagreement as to what it means to claim that Car X's fuel consumption is five times more than Car Y's. It might be that Car X uses 25 gallons of gas and Car Y uses 5 gallons of gas to drive the same distance. But is it true that Car Y's fuel consumption is five times less than Car X's? Five times 25 gallons is 125 gallons, so if Car Y's fuel consumption is five times less than Car X's, it would imply that Car Y uses 125 gallons of gas less than Car X. Once again, this brings Car Y's fuel consumption below zero which is unrealistic.

While it is the second claim in this advertisement that might initially catch one's eye, the first claim also opens up some possibilities for student exploration. How is a carbon footprint measured? What are the units? What does it take to reduce a footprint by 54%? The online calculators can be used as a starting point to investigate these questions.

Assessment

Survival Rates: While a more formal assessment is possible for this example, it may be best used to form the basis for a class discussion on the differences between odds and probability and perhaps the notion of conditional probability. The figures used in this example may be used later for individual in-class assessment. For example, a test item may ask students to use correct language to state the odds of surviving an airplane crash if the probability of survival is assumed to be 24%.

Automobile Trade-in Value: Students turn in a written analysis of their findings. The written report should 1) be well organized, 2) clearly state all assumptions (e.g. MSRP of a trade-in or a new automobile), 3) examine the effect of older or newer trade-ins, 4) examine the range of possible deductions for any given trade-in model, and 5) offer some personal impression on the overall effect of the advertisement.

Environmental Claim: Having students provide individual written responses to items 1-3 before any class discussion allows the instructor to assess students' prior knowledge and preconceptions regarding how one measures and describes change. The instructor can then address any common misconceptions in a subsequent class. Item 4 can be used as the basis for a more extended project if desired.